The MLE Solution for the Location Parameter of the 2-Parameter Exponential
Distribution

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

In Weibull++, when using
the 2-parameter exponential distribution, the
software always sets the location
parameter, gamma, equal to the first
time-to-failure. Is there any theoretical explanation for it? In this article
we will explain this special case of the MLE solution for the 2-parameter
exponential distribution.

Maximum Likelihood Estimation Method

The general likelihood function is given by:

where:

f(x;θ1,θ2,...,θk), is the pdf of a continuous random variable x.

θ1,θ2,...,θk are k unknown parameters that need to be estimated.

x1,x2,...,xR
are R independent observations, which correspond to failure times in life
data analysis.

For the 2-parameter exponential distribution, the log-likelihood function is
given as:

To find the pair
solution , the
equations and
have to be solved.

and

Now let us first examine Eqn. (5). To get the MLE solution
for γ, Eqn. (5) has to be set to zero. We can see
that Eqn. (5) is satisfied if and only if:

or:

However, Eqn. (6) cannot be the solution, because we want to maximize
Eqn. (3); and Eqn. (5) could not be achieved because the summation of failure
times will not be zero for the common cases. This indicates there is no
non-trivial MLE solution that satisfies
both and
.

To solve this dilemma, let us look at the effect
of γ on the exponential distribution. The location
parameter γ, if positive, shifts the beginning of
the distribution by a distance of γ to the right
of the origin, signifying that failures start to occur only
after γ hours of operation and could not occur
before, which means γ ≤ T1 (the
first failure time). As an example, Figure 1 displays the effect
of γ on the exponential distribution with
parameters (λ = 0.001,
γ = 500)
and (λ = 0.001,
γ = 0). The physical meaning
of γ has shed the light on solving this
2-parameter exponential distribution using the MLE method. With the
failure data, the partial derivative Eqn. (5) will be greater than zero. This
implies that the likelihood function in Eqn. (2) increases
when γ is changed from 0
to T1. (For a continuous
function g(x), if its first derivative
is always greater than 0 during the interval [a, b],
then g(x) will increase in the
interval [a, b].) To get the maximum likelihood estimators, we need to
maximize Eqn. (2) so γ should be set
to T1, the first time-to-failure. And
then find λ such
that .

Figure 1: The effect of the location parameter on the
exponential distribution

Example

A reliability engineer conducted a reliability test on 14 units
and obtained the following data set. From the previous testing
experience, the engineer knew that the data were supposed to follow
a 2-parameter exponential distribution.

Table 1 - Life Test Data

Data point index

Time-to-Failure (days)

1

8

2

10

3

15

4

22

5

25

6

30

7

35

8

40

9

50

10

60

11

68

12

73

13

82

14

90

To get the MLE solution for this data analysis, the reliability engineer
used Weibull++ and entered the data in a standard folio. To
set the analysis to MLE, the engineer
clicked the blue link in the Analysis Settings area
of the control panel, as shown next.

The engineer then clicked the Calculate
icon. As shown in the following picture, the results shows that
λ = 0.0282, and γ = 8.0, indicating
that γ is set equal to the first failure time.

As seen in the following Reliability vs. Time plot, the failures
started only when the time equaled 8 hours.

Figure 4: Reliability vs. Time plot for life test data

Conclusion

In the article, we discussed the MLE solutions for the 2-parameter exponential
distribution. While there is no regular MLE solution for this distribution, the
physical effect of the location parameter, γ, leads
us towards the solution. By setting γ equal to
the first time-to-failure and obtaining λ such
that , a maximum can
be achieved along the λ axis, and a local
maximum along the γ axis
at γ = T1, constrained
by the fact that γ ≤ T1.